Radiophysics and Quantum Electronics, Vot. 36, No. 7, 1993

W A V E - B E A M D I F F R A C T I O N IN S M O O T H L Y I N H O M O G E N E O U S M E D I A

I. G. Kondrat'ev, G. V. Permitin, and A. I. Smirnov

Problems of wave propagation and diffraction in smoothly inhomogeneous media involve very considerable difficulties owing to the lack of analytic methods for the solution of wave equations in partial derivatives with variable coefficients as well as to the cumbersomeness and inefficiency' of numerical methods. There exist several methods for the joint description of refraction and diffraction effects in the vicinities of caustics and focal points that are based on geometrical optics (GO) and are its wave extensions, such as the Kravtsov- Ludwig method of standard functions and the Maslov method. However, these methods are unsuitable for the elimination of integrated (quasidiffusional) disturbances of GO, which lead to the gradual loss not only of its applicability but also its informativeness over extended propagation paths. Unlike diffraction catastrophes, integrated (mild) diffraction effects are undetectable "from within" a GO approximation, which makes them especially dangerous in a short-wave asymptotic field description. In. the absence of refraction, in homogeneous media and in vacuum, conversely, integrated diffraction effects are classical. Fresnel provided a fairly detailed description of these effects at a time when the wave equations were unknown. The Fresnel method is based on the representation of a wave field as a superposition of interfering virtual objects (spherical waves) for which geometrical optics is everywhere valid. This idea of using GO not as an approximation of the unknown wave fields but for calculation of the parameters of their integral expansions has gained undergone development in the diffraction theory of aberrations of optical lens systems [1]. An extension of aberration theory to smoothly inhomogeneous media has been proposed [2]. A wave-beam field has been represented as an expansion in short-wave asymptotics of functions of a point source in a small-angle approximation [3]. An approximate mapping of a beam field in vacuum into a beam field in an inhomogeneous medium has been obtained [4]. Concepts of characteristic diffraction zones have been extended to smoothly inhomogeneous media [5]. Here we shall employ a unified approach to consolidate the earlier results [2-5]. 1. QUASIOFrlCS OF SMOOTHLY INHOMOGENEOUS MEDIA We shall examine the problem of diffraction in a smoothly inhomogeneous medium of a monochromatic wave beam U ( r ) e x p ( - i ~ t ) whose behavior is described by a scalar Helmholtz equation:

AU + k2~(~U = 0.

(1)

It is assumed that everywhere on the propagation path the beam remains narrowly directional and has transverse dimensions A that are small in comparison with the characteristic inhomogeneity scale of the medium L~ - e/[Ve 1- In other words, the problem has two small parameters: the characteristic width of the angular spectrum of the beam ~, and the ratio/~ = A/L~, Our aim is to construct solutions of Eq. (I) that are asymptotic in parameters v and #. It can be shown (and this follows from the theory presented below) that for ~, < < 1 and # > > 1 wave beams are propagated along a geometrical-optics ray that emerges from the center of the emitting aperture in the direction of the maximum of its directivity pattern. The equation of this ray (we shall call it the "reference" ray) has the form

Institute of Applied Physics, Russian Academy of Sciences. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Radiofizika, Vol. 36, No. 7, pp. 616-622, July, 1993. Original article submitted October 19, 1992.

394

0033-8443/93/3607-0394512.50

9

Plenum Publishing Corporation

lye

d2f'o

dr 2 ~o(r = 0) = 0,

= 2

'

(2)

~0(r = 0) = ~/~(0)Zo.

Here, r0&) is the radius vector of points on the reference ray as read from the center of the emitting aperture; z o is a unit vector in the direction of the directivity-pattern maximum of the emitting aperture; and the variable ~- is related to the arc length of the reference ray s as dr = ds/e~ro). For a description of the beam-field structure, it is convenient to convert to a orthogonal curvilinear coordinate system (T, ~, ~/) that is associated with the reference ray:

(3) where ( l , a, b) is an orthogonal basis that has been turned with respect to the natural triangle ( l , n, m) by the angle 0(~-), which varies along the reference ray in accordance with Rytov's law:* 0 = 6 = ,]~/T, where T is the radius of torsion of the reference ray. The Larn6 coefficients or the curvilinear coordinate system are

O where p is the principal radius of curvature of the reference ray. We represent the field U as ~r

U(r,~,rl) = (eo(r)) -1/4. W(r,~,rl) e x p ( z k / c o ( r ) dr),

(4)

o

where W is the smoothly (in the scale of X) varying field amplitude, and %(r) is the dielectric constant on the reference ray. Substituting (4) into (1) and ignoring terms on the order of u4 and/zv 3, we obtain the following shortened equation:

(5)

Equation (5) is valid in a quasioptical approximation for an arbitrary three-dimensionally inhomogeneous medium. Below, however, we shall restrict our examination to two-dimensionally inhomogeneous fields and media (0/03, = 0/8~? = O) to avoid overloading the text with cumbersome computations. Retaining in Eq. (5) only the terms that are cubic in v and/z, we obtain

2zk~rW + 02

-

-

- \ ~0~2 +k2t~3(r)~3w =0,

[()* 31

9 3(,-)=

-

a, ,__a,,]

(6)

4,2

*The law was established for rotation of the polarization plane of electromagnetic radiation in an approximation of geometrical optics. 395

Here, 0 is the radius of curvature of the reference ray:

Equation (6) is written in a form that allows a two-dimensionally inhomogeneous medium to be associated with an equivalent optical line of distributed quadratic and cubic phase correctors. The absence of linear correctors (prisms) in Eq. (6) indicates that the beam is localized in the vicinity of the reference ray. In the equivalent optical line, the quadratic correctors correspond to ideal lenses, but cubic (and higher-order) corrections result in aberrations. 2. NONABERRATIONAL APPROXIMATION If aberrations are ignored in Eq. (6) (leaving only terms that are quadratic in ~), we obtain the equation of quasioptics in an ideal lens-like medium [6]:

22kOW~

02W~ k 2 , ~ ( r ) ( ~ W ~ = 0,

(7)

--gT- + 0(-----V where W~ is the complex field amplitude in a nonaberrational approximation. A solution of Eq. (7) with arbitrary "initial" conditions (W~(z = 0, ~) = W0(~)) can be written in quadratures if we know the fundamental system (Ol, a2) of the equation of the rays that are differentially close to the reference ray:

+ ~2(r)o. = O; (8)

o.1(0) = 1, ~1(0) = o; o.2(0) = 0, ~ ( 0 ) = 0. The field in a nonaberrational approximation has the form:

w~ = f Wo(<)a,(r, (, r a<, (9)

[o1(,

Gs =

~,

,k t2o.~

o.i + ~

( ~ - <

' '

where G~ is Green's function for Eq. (7). Direct comparison of G~ with Green's function for the quasioptical equation in a homogeneous medium (vacuum),

shows that, within a framework of the nonaberrational approximation, a beam field in vacuum is mapped into a beam field in a smoothly inhomogeneous medium by means of the substitution z --,/~ = o.2/o.1,

l

w~(~,() = --~_~ ~

396

x --, (/o'1,

(,k~-~al,') w
(11)

where W(~ x) is the smoothly varying beam-field amplitude in vacuum with the same "initial" distribution W0(x); and R(r) is the radius of curvature of the phase front of the wave, which diverges from point (r, ~ = 0), at the center of the emitting aperture. The transformation of coordinates and fields (11)-~ that has been performed allows the concepts of the zone of geometrical optics and the characteristic Fresnel and Fraunhofer diffraction zones to be extended to smoothly inhomogeneous media. The role of the Fresnel parameter in this case is played by the quantity p = x/T~ k0/A0, where A0 is the beam width in the plane r = 0. Unlike wave diffraction in Vacuum, where the Fresnel parameter increases monotonically with distance from the initial aperture, in smoothly inhomogeneous media it is in general a nonmonotonic function of ~-. Since the order of alternation of the characteristic diffraction zones in inhomogeneous media can differ greatly from the classical order, GO zone (near, intermediate) --- Fresnel zone --- Fraunhofer zone, which occurs when a plane wave is diffracted by a slot in a shield. In defocusing media (q'2 < 0), the GO region (p < < 1) can extend from the emitting aperture to infinity. In focusing systems,~: (if2 > 0), conversely, the zone alternation is accelerated, and a Fraunhofer diffraction pattern is established at a finite distance in the vicinities of "focal planes," where p --, co (i.e., near points at which the reference ray touches the caustics of the initially plane wave). Then, as a rule, the zone order is reversed, and a GO region (p < < 1) is again established near the caustic of the field of a point source located at the center of the emitting aperture. By analogy with lens systems, the beam cross-sections in which the GO zone is restored can be called image planes (they are sometimes called zones of convergence). The described change of the directions of ordering of the characteristic diffraction zones can, in principle, occur many times. Any of the characteristic zones - not necessarily the Fraunhofer zone - can be established at infinity (r --, co). It should be noted that, unlike in optical lens systems, whose elements are fairly close to ideal correctors, a nonaberrational approximation has very limited applicability in arbitrary smoothly inhomogeneous media, and aberrations lead not to corrections but to substantial changes of field structure. 3. ABERRATIONAL DISTORTIONS OF BEAM FIELD A method based on the representation of a wave field at the system output in the form of a convolution of an ideal "image" with the so-called transmission function has been developed in the diffraction theory of discrete optical systems [1]. Extending this method to smoothly inhomogeneous media, we represent the complex amplitude of the field near the reference ray as oo

= fw,(r,c) K(e_ r (12) The kernel of the integral transformation K(X, r, ~) will be called the transmission function of a smoothly inhomogeneous medium; for small aberrations, it is narrowly localized with respect to the variable X near zero and is smooth with respect to variables r and ~. The transmission function has a very obvious physical meaning; it describes the field in the cross-section r = const created by sources distributed in the plane r = 0 that have a constant amplitude and a quadratic phase dependence, which ensures focusing on the point (r, ~ + X). In turn, Eq. (12) is an expansion in such virtual waves of the field of an arbitrary wave beam.** Along with the transmission function, the diffraction theory of aberrations contains a transmission coefficient, which is its Fourier transform: tin fact, Eq. (11) is an extension of the well-known [7, 4] transformation of an ideal thin lens to smoothly inhomogeneous media. :)As is apparent from Eq. (6), any ray curvature is a focusing factor. In addition, layers with a refractive-index maximum are focusing. **System of fimctions K is complete and orthogonal. In fact, if we place in the cross-section r = 0 two mutually compensating ideal quadratic phase correctors and in the space between them expand the field in plane waves, we obtain expansion (12) at the out-put. In special cases, for beam cross-sections in which o2(r) = 0 or al(r ) = 0, (12) becomes an expansion in functions of a point source or in initially plane waves. 397

r'q

t.r,

398

r,,) = f K(x,.,,) exp(,kXa)d X

9

(13)

--OO

Applying convolution theory to Eq. (12), we obtain another representation for the beam field: (14) where W~(z, K) is the Fourier transform of the beam field in a nonaberrational approximation. Details on calculation of: transmission functions and coefficients were presented earlier [2]; here we shall present formulas for them for cases in which fourth-order (and higher) aberrations are insignificant: K = exp

-tk X

K(X,r,5) = ~

a,n~ma a-m

;

(15)

m----0

1

V (Ca:+ ~2)/Aab) exp (,k~o),

(16)

~, = ((a~f27ag) - ~3) 5a + (all3aolSCx + ~f'); where V(x) is the Airy function, /3 = a 2 - a2/3ao, and Aab = (3ao/k2)l/3 is the characteristic aberrational scale. The parameters am(r) in Eqs. (t5) and (16) satisfy the system of ordinary differential equations ~0 = a l

dl = 2a2 - 3~2ao - l / e ,

ti~ = 3as - 2q2al,

aa = - ~ 2 a 2 + 0,5r

(17)

and initial conditions am(r = O) = O. As can be seen from Eqs. (12)-(16), beam structure depends on a rather large number of parameters. However, the main parameter determining the importance of aberrational distortions is the ratio Aab/A~, where, we recall, A~ is the beam width in a nonaberrational approximation. The two following limiting situations can be distinguished:

~)A, > tAobt : W(,-,f)-~ Ws0",f + a f 2) Cxp (-,J:a3(~-)f3).

(18)

Aberrational distortions in such regions affect only the amplitude and phase structures of the peripheral part of the beam, where the field amplitude is Iow. Near the reference ray, the field is fairly well-described in a nonaberrational approximation. It must be noted that expression (18) is valid for single-scale (bell-shaped) distributions of W~ or in those cases in which all characteristic scales of W~ are much greater than Aab. W ( r , r = (1/v~A~b)vC(/Aab) ~"s(r, 0).

(19)

At this limit, cubic aberrations lead to the formation of an Airy field structure near the reference ray that are typical of the vicinities of a smooth caustic. However, formula (19) does not describe correctly the asymptotic behavior of the field at great distances from the reference ray (where it is weak) - the law of decay is determined not by the asymptotic behavior of the Airy function but by the angular spectrum of the beam in a nonaberrational approximation.

399

A graphics program was written for calculation of the fields of wave beams in i~momogeneous media; the calculation results (for the special case of a parabolic dielectric-constant layer) are presented in Figs. 1 and 2. The reference ray, the beam boundaries at the 1/e decay level, and the structure of field intensity in several characteristic cross-sections in a nonaberrational approximation are shown in Fig. 1. In Fig. 2, the same items are shown with allowance for strong cubic aberrations. The authors thank A. L. Sharova and A. V. Lipatov for assistance in writing the program. REFERENCES 1. 2. 3. 4. 5. 6. 7.

400

M. Born and E. Wolf, Fundamentals of Optics [Russian translation], Nauk_a, Moscow (1970). I . G . Kondrat'ev, G. S. Permitin, and A. I. Smirnov, Izv. Vyssh. Uchebn. Zaved., Radiofiz., 23, No. 10, 1195 (1980). G.V. Permitin, Dissertation, Gorky (1973). G.V. Permitin, Izv. Vyssh. Uchebn. Zaved., Radiofiz., 16, No. 2, 254 (1973). I . G . Kondrat'ev and G. V. Permitin, Proceedings of Fifth Colloquium on Microwave Communication, Budapest (1974). V.I. Talanov, Dissertation, Gorky (1967). V.I. Talanov, Pis'ma Zh. I~ksp. Teor. Fiz., 11,303 (1970).